You are going to learn the skills to take real life data and create a mathematical model which explains what is happening, which you can then use to make further predictions. This is an important skill for business, economics, insurance etc. However it is not a "black and white" process - models are approximations. You have to decide which model is the most accurate, and you must also recognise how your model is limited.
Preparatory Skills
1. Identifying shapes of data
Each of these graphs show two variables plotted on a scatter graph. State whether each one is linear, quadratic or cubic.
Look again at the data. How would you describe the relationship between the variables? Discuss this in your class.
3. Modelling the data more exactly
Once you have established that a relationship between two variables exists, we need to find the equation of the line of best fit.
In the case of a linear relationship, this is relatively straight forward, and was covered in IGCSE. Click HERE to recap.
There are three techniques for finding the equation of a quadratic graph. The first is using your knowledge of transformations and a little bit of "guesstimation". The second is more exact and uses algebra and simultaneous equations. The third is demonstrated in mymaths and allows you to reduce a quadratic variable to a linear one.
Modelling by eye
Use Graph 2 for this exercise, which is available HERE.
We can see that this data is quadratic, i.e. y = ax2 + bx + c. We will start with the graph y = x2 and transform it to fit our data.
We'll start by moving the vertex/base of the graph. It needs to be roughly where the pink cross is, i.e. moving from (0,0) to about (13, 1). That is, the whole graph needs to move 13 to the right and 1 up.
Think back to your graph transformations...
To transform f(x) to the right by a, do f(x - a)
To transform f(x) up by 1, do f(x) + a.
So, plot the graph y = (x - 13)2+ 1. Press INSERT, and type (x - 13)^2 + 1.
So, the base is in the right place, which we have achieved using transformations of graphs. Now we need to "squash" it in the y direction. So, experiment by putting a decimal, such as 0.5, in front of the equation. 0.5(x - 13)^2 + 1. Try some different decimals until you get the best fit. (To change the graph, double click on the equation on the left).
Once you have found a good line, expand the bracket to give an equation in the form y = ax2 + bx + c.
Check this again to make sure it gives you the right line.
Deducing the line of best fit using algebra
This technique ues simultaneous equations to deduce a line of best fit. You need to choose three points from your graph that a quadratic curve will pass through. In this example, we will use (6,9), (12,1) and (18,4).
We know that a quadratic equation is of the form
y = ax2 + bx + c
So, substituting in the above values gives us three equations:
9 = 36a + 6b + c
1 = 144a + 12b + c
4 = 324a + 18b + c
These can be solved simultaneously or using a graphical calculator.
Plotting one variable against the square of another variable
This is another technique you can use, demonstrated nicely in mymaths here. (Choose number 3 on the side)
4. Using your model
Once you have worked out a model, you can use it to predict the value of one variable from another. Take this graph which shows the relationship between the mass on the end of a spring and the length of the spring. (Hooke's Law).
If you put a mass of 350g, what length would you expect to see?
If the spring was 29cm long, what mass you would expect to see?
It is most important that you can answer these questions:
Is my model a reasonable explanation of what is going on?
How accurate are the results I get from my model?
What can my model NOT tell me?
How could I further develop my model/investigation?
In relation to the weights on a spring example above:
Is my model a reasonable explanation of what's going on? Yes. As you put more weights on the spring, it gets longer. That's what I'd expect. I'd also expect it to be linear, i.e. each extra 100g increases the length by the same amount. (Although, would that ALWAYS be the case? Think about what would happen for really big weights).
How accurate are the results I can get from my model? It looks like I have measured weight to the nearest 100g and length to the nearest 0.1cm. So I can't expect my results to be any more accurate than that.
What can my model NOT tell me? The largest weight I used was 500g. If I wanted to predict what would happen for 1000g, I can't be sure that the relationship will hold. For example, the spring might get distorted (or break!) if the weight gets much bigger than 500g. Eventually it would fully extend, looking like a piece of wire, so that adding extra weights would make no difference.
How could I further develop my investigation? Answer this based on the last two questions... so, I could make my measurements more accurate. I could use a different technique to choose the model (for example, if I used the "by eye/guesstimation" method above, I might try the algebraic one instead.) I might get some results using larger weights. I might also think about different springs, made of different materials, in different room temperatures etc. etc.
You could also look for new data on the internet. You don't have to collect it yourself. But say why that new data might not be trustworthy.
These answers above are short and are just here to give you an idea of the sorts of things you need to discuss. Your investigation should be far more detailed.
6. More Practice
Three different sets of data are given below. Using the techinques above:
1. Write down the variables you have been given. State the range of each, and the accuracy they have been given to.
2. Model the data by transforming a simpler graph (say, y = x^2)
3. State whether or not this fits the expectations, and why.
4. State how well the data fits the model (i.e. how well the points match the curve)
5. Remodel the graph by choosing points on the graph and using an algebraic method. Compare this to your initial model - does it fit better?
6. Show what data you can find from your model.
7. State what data you CANNOT find from your model.
8. How would you improve your model? Could it be applied to other, similar situations?
Further Help
There is a good demonstration on www.mymaths.co.uk that will take you through the modelling process step by step. Working through this lesson, and then doing the online homework, will prepare you well for the real portfolio task.
Answers
1. Identifying shapes of data
.
The relationships between:
Weight and height are linear: y = mx + c
Test score and hours of revision are quadratic: y = ax2+ bx + c
Top score on Tekken and hours of practice are cubic: ax3 + bx2+ cx + d
Hours studying and hours watching TV are linear: y = mx + c
No. of friends on Facebook and Time spent in the gym are quadratic: y = ax2+ bx + c
X and Y are quadratic: y = ax2+ bx + c
4. Using your model
If you put a mass of 350g, what length would you expect to see? About 38cm.
If the spring was 29cm long, what mass you would expect to see? About 150g.
Mathematical Modelling
Introduction
You are going to learn the skills to take real life data and create a mathematical model which explains what is happening, which you can then use to make further predictions. This is an important skill for business, economics, insurance etc. However it is not a "black and white" process - models are approximations. You have to decide which model is the most accurate, and you must also recognise how your model is limited.
Preparatory Skills
1. Identifying shapes of data
Each of these graphs show two variables plotted on a scatter graph. State whether each one is linear, quadratic or cubic.Click here for some answers
2. Describing the data in real life
Look again at the data. How would you describe the relationship between the variables? Discuss this in your class.3. Modelling the data more exactly
Once you have established that a relationship between two variables exists, we need to find the equation of the line of best fit.In the case of a linear relationship, this is relatively straight forward, and was covered in IGCSE. Click HERE to recap.
There are three techniques for finding the equation of a quadratic graph. The first is using your knowledge of transformations and a little bit of "guesstimation". The second is more exact and uses algebra and simultaneous equations. The third is demonstrated in mymaths and allows you to reduce a quadratic variable to a linear one.
Modelling by eye
Use Graph 2 for this exercise, which is available HERE.Open this file: _modelling_data.xls Select and copy the data set.
Open Graph. Click Insert Point Series.
We can see that this data is quadratic, i.e. y = ax2 + bx + c. We will start with the graph y = x2 and transform it to fit our data.
We'll start by moving the vertex/base of the graph. It needs to be roughly where the pink cross is, i.e. moving from (0,0) to about (13, 1). That is, the whole graph needs to move 13 to the right and 1 up.
Think back to your graph transformations...
To transform f(x) to the right by a, do f(x - a)
To transform f(x) up by 1, do f(x) + a.
f(x) = x2.
f(x - 13) = (x - 13)2
f(x - 13) + 1 = (x - 13)2
+ 1.
So, plot the graph y = (x - 13)2+ 1. Press INSERT, and type (x - 13)^2 + 1.
So, the base is in the right place, which we have achieved using transformations of graphs. Now we need to "squash" it in the y direction. So, experiment by putting a decimal, such as 0.5, in front of the equation. 0.5(x - 13)^2 + 1. Try some different decimals until you get the best fit. (To change the graph, double click on the equation on the left).
Once you have found a good line, expand the bracket to give an equation in the form y = ax2 + bx + c.
0.1(x - 13)2+ 1
0.1(x2- 26x + 169) + 1
0.1x2- 2.6x + 16.9 + 1
0.1x2- 2.6x + 17.9
Check this again to make sure it gives you the right line.
Deducing the line of best fit using algebra
This technique ues simultaneous equations to deduce a line of best fit. You need to choose three points from your graph that a quadratic curve will pass through. In this example, we will use (6,9), (12,1) and (18,4).
We know that a quadratic equation is of the form
y = ax2 + bx + c
So, substituting in the above values gives us three equations:
9 = 36a + 6b + c
1 = 144a + 12b + c
4 = 324a + 18b + c
These can be solved simultaneously or using a graphical calculator.
Plotting one variable against the square of another variable
This is another technique you can use, demonstrated nicely in mymaths here. (Choose number 3 on the side)
4. Using your model
Once you have worked out a model, you can use it to predict the value of one variable from another. Take this graph which shows the relationship between the mass on the end of a spring and the length of the spring. (Hooke's Law).
If you put a mass of 350g, what length would you expect to see?
If the spring was 29cm long, what mass you would expect to see?
Answers here.
5. Limitations of your model
It is most important that you can answer these questions:In relation to the weights on a spring example above:
Is my model a reasonable explanation of what's going on? Yes. As you put more weights on the spring, it gets longer. That's what I'd expect. I'd also expect it to be linear, i.e. each extra 100g increases the length by the same amount. (Although, would that ALWAYS be the case? Think about what would happen for really big weights).
How accurate are the results I can get from my model? It looks like I have measured weight to the nearest 100g and length to the nearest 0.1cm. So I can't expect my results to be any more accurate than that.
What can my model NOT tell me? The largest weight I used was 500g. If I wanted to predict what would happen for 1000g, I can't be sure that the relationship will hold. For example, the spring might get distorted (or break!) if the weight gets much bigger than 500g. Eventually it would fully extend, looking like a piece of wire, so that adding extra weights would make no difference.
How could I further develop my investigation? Answer this based on the last two questions... so, I could make my measurements more accurate. I could use a different technique to choose the model (for example, if I used the "by eye/guesstimation" method above, I might try the algebraic one instead.) I might get some results using larger weights. I might also think about different springs, made of different materials, in different room temperatures etc. etc.
You could also look for new data on the internet. You don't have to collect it yourself. But say why that new data might not be trustworthy.
These answers above are short and are just here to give you an idea of the sorts of things you need to discuss. Your investigation should be far more detailed.
6. More Practice
Three different sets of data are given below. Using the techinques above:1. Write down the variables you have been given. State the range of each, and the accuracy they have been given to.
2. Model the data by transforming a simpler graph (say, y = x^2)
3. State whether or not this fits the expectations, and why.
4. State how well the data fits the model (i.e. how well the points match the curve)
5. Remodel the graph by choosing points on the graph and using an algebraic method. Compare this to your initial model - does it fit better?
6. Show what data you can find from your model.
7. State what data you CANNOT find from your model.
8. How would you improve your model? Could it be applied to other, similar situations?
Further Help
There is a good demonstration on www.mymaths.co.uk that will take you through the modelling process step by step. Working through this lesson, and then doing the online homework, will prepare you well for the real portfolio task.Answers
1. Identifying shapes of data
.The relationships between:
Weight and height are linear: y = mx + c
Test score and hours of revision are quadratic: y = ax2+ bx + c
Top score on Tekken and hours of practice are cubic: ax3 + bx2+ cx + d
Hours studying and hours watching TV are linear: y = mx + c
No. of friends on Facebook and Time spent in the gym are quadratic: y = ax2+ bx + c
X and Y are quadratic: y = ax2+ bx + c
4. Using your model
If you put a mass of 350g, what length would you expect to see? About 38cm.
If the spring was 29cm long, what mass you would expect to see? About 150g.